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PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
CL
w
A
Fig.130   Schematicillustration ofinduced drag.
Fig.131    Variation oflift coeffiaent with angle of attack for finite w:ings.
REVIEW OF BASlC AERODYNAMIC PRINCIPLES
31
of attack for a given lift coefficient is higher than that for a two-dimensional wing
(A = oo). In other words, the lift-curve slope effectively reduces with a decrease
in aspect ratio. Based on the lifting line theory, we can obtain an expression for
the lift-curve slope of the fi:tute wings as follows.
    For a given lift coefficient,
CL - aoao
     = a(LYo + at)
      = aa'o (1+ -..)
(1.49)
(1.50)
(1.52)
where a(, is the sectional or two-dimensionallift-curve slope and a is the lift-curve
slope of a finite wing of aspect ratio A. In the above formula, the values of a and
ao are per rad.
1.7  Methods of Reducing Induced Drag
      An obvious method of reducing induced drag is to increase the aspect ratios of a
lifting surface or install end plates (at the wing tips) to pre'vent the crossflow around
wing tips. However, either of these two approaches is not always the best option
from structural considerations. The weight of'the wing increases considerably
because of the additional structure that is needed to resist the large bending loads
caused by increased span or the added weight ofthe end plates.Another met,hod that
is simpler and does not encounter this problem is the use of winglets as discu,ssed
in the following section.
    One effective method of reducing induced drag of a lifting surface is the ap-
plication of winglets. A winglet is a small "wing" placed near the wing tips and
almost normal to the main wing surface as shown in Fig. 1.32. Placing a winglet
on an existing wing alters the spanwise distribution of the circulation, hence the
structure of the wing tip vortices. The key to achieving induced-drag reduction
is the efficient generation of a side force. The side force is generated due to the
lift-induced inflow above the wing tip or due to the outfiow below the wing tip.
The side force dyN generated by the winglet because of fiow angle a' is diret:ted
inboard as shown in Fig. 1.32b. This side force provides a component -dD in
the fiight direction that contributes to a reduction in induced drag. This effect is
similar to that on a sail of a sailboat tacking up wind.
    The application of winglets to KC135 aerial~tanker aircraft resulted in approx-
imately 9% reduction in drag at cruise conditions. Winglets are also used on the
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32               PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
                    .         'vi
Wing
a) Schematic flow o'ver winglet
      b) Side force on winglet
Fig.132 Conceptofwinglets.
Boeing 747-400 aircraft, which is a long-range derivative of the Boeing 747 wide-
body transport family.
1.8   Tip Vortices: Formation and Hazards
       A salient feature offinite wingsis the existence of tip vortices and their influence
on the aircraft that fly in their vicinity. Note that for a two-dimensional wing, which
is supposed to extend from -oo to oo, there are no tip effects and tip vortices do
not exist.
   Tip vortices on a finite wing are mainly caused by the pressure differences ex-
isting on the upper and lower surfaces of the wing. For a lifting wing operating
at positive angles of attack, the pressure on the upper surface is lower compared
to that on the lower surface. Be9ause of this pressure differential, there is"~ ten-
dency for the fluid to "leak" from lower surfaces to higher surfaces, and wing
tips provide a path to this "leakage:' Thus, fluid particles from the lower surface
tend to go around the wing tips and move on to the upper surface, forming a
curved, vortex type of fiow as shown in Fig. 1.33a. This curved fiow, combined
with the freestream airflow, when viewed from top, appears to be moving in a
helical path as shown in Fig. 1.33b. A short distance downstream of the wing,
　
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